Spectral Methods

These examples are based on the book

L. N. Trefethen, Spectral Methods in MATLAB, SIAM.

I have rewritten these codes in Python and added more examples.

I have some slides which I use for teaching along with this code, get them here.

You can get the codes and run them on your own computer

git clone https://github.com/cpraveen/chebpy
cd chebpy
jupyter-lab

For nbviewer links to these files, see here

https://cpraveen.github.io/teaching/chebpy.html

Run the code in binder:

Run the code in colab:

Matlab versions of the code are available here

https://github.com/cpraveen/spectral_matlab

Table of contents¶

• Spectral Methods
• p01: Convergence of fourth order finite differences
• p02: Convergence of periodic spectral method
• p03: Band-limited interpolation
• p04: Periodic spectral differentiation
• p05: Repetition of p04 via FFT
• p05b: Repetition of p05 via real FFT
• p05c: Repetition of p05 using scipy.fftpack.diff
• p06: Variable coefficient wave equation
• Variable coefficient acoustics equation
• p07: Accuracy of periodic spectral differentiation
• p08: Eigenvalues of harmonic oscillator
• p09: Polynomial interpolation in equispaced and chebyshev points
• p10: Polynomials and corresponding equipotential curves
• p11: Chebyshev differentiation of a smooth function
• p12: Accuracy of Chebyshev spectral differentiation
• p13: Solve linear BVP
• p14: Solve nonlinear BVP
• p15: Solve eigenvalue BVP
• p16: Poisson equation in 2-D
• p17: Helmholtz Equation in 2-D
• p18: Chebyshev differentiation via FFT
• p19: Second order Wave Equation on Chebyshev Grid
• p20: Second order wave equation in 2-D via FFT
• p21: Eigenvalues of Mathieu operator
• p22: 5’th eigenvector of Airy equation
• p23: Eigenvalues of Laplacian in 2-D
• p23a: Eigenvalues of perturbed Laplacian in 2-D
• p24: Pseudospectra of Davies’s complex harmonic oscillator
• p25: Stability regions for ODE formulas
• p26: Eigenvalues of 2nd order Chebyshev diff. matrix
• p27: Solve KdV equation
• Linear advection equation using FFT and RK4
• Burgers equation using FFT
• Eigenvalues of Chebyshev differentiation matrix
• p30: Spectral integration, ODE style
• p32: Solve linear BVP
• p33: Solve linear BVP, Neumann bc